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JetCalculus[EvolutionaryVector] - form the evolutionary part of a vector field
Calling Sequences
EvolutionaryVector(X)
Parameters
X - a vector field or a generalized vector field on a bundle E -> M
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Description
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The total part (TotalVector) of a generalized vector field X on the bundle E -> M is the generalized vector field Y on E -> M such that X - Y is a vertical vector and Hook(Y, omega) = 0 for any contact 1-form omega on J^1(E).
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The vertical vector X - Y is called the evolutionary part of the vector field X.
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The evolutionary part of a projectable vector field X has the following geometric interpretation. Let Phi _t: E -> E be the flow of X. Then Phi_t covers a map Psi_t: M -> M. If sigma: M -> E is a section of E, then the induced flow in the space of sections is defined to be sigma_t(x) = Phi_t (sigma(Psi_( - t)(x))). The derivative of sigma_t with respect to t, evaluated at t = 0, yields the components of the evolutionary part of X.
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The command EvolutionaryVector is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form EvolutionaryVector(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-EvolutionaryVector(...).
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Examples
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Example 1.
Create a space of 2 independent variables and 2 dependent variables.
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![DGsetup([x, y], [u, v], J22, 1)](/support/helpjp/helpview.aspx?si=6555/file05755/math107.png)
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Define a vector X1 and compute its total and evolutionary parts totX1 and evolX1. Check that X1 = totX1 + evolX1.
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J22 >
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| (2.1) |
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J22 >
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![totX1 := D_x+u[1]*D_u[]+v[1]*D_v[]](/support/helpjp/helpview.aspx?si=6555/file05755/math133.png)
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J22 >
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![evolX1 := -u[1]*D_u[]-v[1]*D_v[]](/support/helpjp/helpview.aspx?si=6555/file05755/math140.png)
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J22 >
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| (2.4) |
Define a vector X2 and compute its total and evolutionary parts totX2 and evolX2. Check that X2 = totX2 + evolX2.
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J22 >
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![X2 := D_u[]](/support/helpjp/helpview.aspx?si=6555/file05755/math163.png)
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![X2 := D_u[]](/support/helpjp/helpview.aspx?si=6555/file05755/math166.png)
| (2.5) |
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J22 >
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| (2.6) |
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J22 >
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![evolX2 := D_u[]](/support/helpjp/helpview.aspx?si=6555/file05755/math180.png)
| (2.7) |
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J22 >
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![D_u[]](/support/helpjp/helpview.aspx?si=6555/file05755/math187.png)
| (2.8) |
Define a vector X3 and compute its total and evolutionary parts totX3 and evolX3. Check that X3 = totX3 + evolX3.
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J22 >
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![X3 := evalDG(a*D_x+b*D_y+c*D_u[]+d*D_v[])](/support/helpjp/helpview.aspx?si=6555/file05755/math203.png)
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![X3 := a*D_x+b*D_y+c*D_u[]+d*D_v[]](/support/helpjp/helpview.aspx?si=6555/file05755/math206.png)
| (2.9) |
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J22 >
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![totX3 := a*D_x+b*D_y+(b*u[2]+a*u[1])*D_u[]+(b*v[2]+a*v[1])*D_v[]](/support/helpjp/helpview.aspx?si=6555/file05755/math213.png)
| (2.10) |
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J22 >
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![evolX3 := (c-b*u[2]-a*u[1])*D_u[]+(d-b*v[2]-a*v[1])*D_v[]](/support/helpjp/helpview.aspx?si=6555/file05755/math220.png)
| (2.11) |
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J22 >
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![a*D_x+b*D_y+c*D_u[]+d*D_v[]](/support/helpjp/helpview.aspx?si=6555/file05755/math227.png)
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Example 2.
In this example we illustrate the geometric interpretation of the evolutionary part of a projectable vector field.
First define a 3-dimensional bundle E over a two dimensional base. Define the base space M separately.
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J22 >
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![DGsetup([x, y], M)](/support/helpjp/helpview.aspx?si=6555/file05755/math243.png) ![DGsetup([x, y], [u], E, 2)](/support/helpjp/helpview.aspx?si=6555/file05755/math244.png)
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Define a vector field X4 and compute its evolutionary part evolX4. Define the projection Y4 of the vector field X4 onto the base manifold M.
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![X4 := evalDG(-y*D_x+x*D_y+u[]*D_u[])](/support/helpjp/helpview.aspx?si=6555/file05755/math262.png)
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![X4 := -y*D_x+x*D_y+u[]*D_u[]](/support/helpjp/helpview.aspx?si=6555/file05755/math265.png)
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![evolX4 := (u[]-x*u[2]+y*u[1])*D_u[]](/support/helpjp/helpview.aspx?si=6555/file05755/math272.png)
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| (2.15) |
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| (2.16) |
Calculate the flow psi_t of Y4 and the flow Phi_t of X4.
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![psi[t] := eval(Flow(Y4, t), t = -t)](/support/helpjp/helpview.aspx?si=6555/file05755/math302.png)
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![psi[t] := [x = x*cos(t)+y*sin(t), y = -x*sin(t)+y*cos(t)]](/support/helpjp/helpview.aspx?si=6555/file05755/math305.png)
| (2.17) |
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![Phi[t] := Flow(X4, t)](/support/helpjp/helpview.aspx?si=6555/file05755/math309.png)
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![Phi[t] := [x = x*cos(t)-y*sin(t), y = x*sin(t)+y*cos(t), u[] = u[]*exp(t)]](/support/helpjp/helpview.aspx?si=6555/file05755/math312.png)
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Define a section sigma of E sending (x, y) to S(x, y).
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![sigma := Transformation(M, E, [x = x, y = y, u[] = S(x, y)])](/support/helpjp/helpview.aspx?si=6555/file05755/math326.png)
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![sigma := [x = x, y = y, u[] = S(x, y)]](/support/helpjp/helpview.aspx?si=6555/file05755/math329.png)
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Calculate the induced flow on the space of sections.
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![sigma[t] := ComposeTransformations(Phi[t], sigma, psi[t])](/support/helpjp/helpview.aspx?si=6555/file05755/math337.png)
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![sigma[t] := [x = (x*cos(t)+y*sin(t))*cos(t)-(-x*sin(t)+y*cos(t))*sin(t), y = (x*cos(t)+y*sin(t))*sin(t)+(-x*sin(t)+y*cos(t))*cos(t), u[] = S(x*cos(t)+y*sin(t), -x*sin(t)+y*cos(t))*exp(t)]](/support/helpjp/helpview.aspx?si=6555/file05755/math340.png)
| (2.20) |
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![ApplyTransformation(sigma[t], [x, y])](/support/helpjp/helpview.aspx?si=6555/file05755/math344.png)
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![[(x*cos(t)+y*sin(t))*cos(t)-(-x*sin(t)+y*cos(t))*sin(t), (x*cos(t)+y*sin(t))*sin(t)+(-x*sin(t)+y*cos(t))*cos(t), S(x*cos(t)+y*sin(t), -x*sin(t)+y*cos(t))*exp(t)]](/support/helpjp/helpview.aspx?si=6555/file05755/math347.png)
| (2.21) |
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)(x, y)*y-(D[2](S))(x, y)*x+S(x, y)]](/support/helpjp/helpview.aspx?si=6555/file05755/math354.png)
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Compare with the components of evolX4.
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![GetComponents(evolX4, [D_x, D_y, D_u[]])](/support/helpjp/helpview.aspx?si=6555/file05755/math364.png)
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![[0, 0, u[]-x*u[2]+y*u[1]]](/support/helpjp/helpview.aspx?si=6555/file05755/math367.png)
| (2.23) |
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